Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

learn more… | top users | synonyms (1)

6
votes
1answer
192 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
6
votes
2answers
321 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
6
votes
3answers
217 views

Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID

In this other post, Qiaochu Yuan comments that the solutions for the homegeneous linear differential equation with constant coefficients are a special case of the structure theorem for finitely ...
5
votes
2answers
329 views

The Green’s function of the boundary value problem

What is the Green’s function of the boundary value problem $$ \frac{\mathrm d^2 y}{\mathrm d x^2}-\frac{1}{x}\frac{\mathrm dy}{\mathrm dx}=1,\quad y(0)=y(1)=0, $$ this boundary problem is not self ...
4
votes
1answer
3k views

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ...
4
votes
6answers
1k views

ODE introduction textbook

Unfortunately I have reached the maximum number of math classes I can take for my undergraduate degree. I still wish to study basic ODEs and basic number theory. What is a good textbook with an ...
3
votes
1answer
33 views

Uniqueness of solutions to linear recurrence relations

I understand that if I have a linear homogeneous recurrence relation of the form $q_n = c_1 q_{n-1} + c_2 q_{n-2} + \cdots + c_d q_{n-d}$, I can construct the characteristic polynomial $p(t) = t^d - ...
2
votes
2answers
389 views

How to solve this ODE?

For a certain problem, I am trying to solve the ODE $$\ddot{z}(t) - \omega^2 z(t) = f_0 \Big(e^{-i(\omega+\delta)t}+e^{-i(\omega-\delta)t}\Big)$$ where $\omega$ is a real and $\delta$ is close to ...
0
votes
1answer
103 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
8
votes
2answers
198 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
8
votes
1answer
402 views

To get addition formula of $\tan (x)$ via analytic methods

Assume that we only know $\tan (0)=0$ and also given the relation $\tan'(x)=1+\tan^2(x)$ about $\tan (x)$ and we do not know other $\tan (x)$ relations of trigonometry. How can I get the additon ...
8
votes
1answer
163 views

Solving the differential equation $\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$

How do we solve the differential equation $$\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$$? I tried substituting $v=yx$ but I do not seem to be getting anywhere.Putting $u=x-2y$ yielded nothing better. ...
7
votes
1answer
121 views

Periodic solution of differential equation y′=f(y)

Let $f∈C^∞(ℝ^2,ℝ^2)$ and $∀x∈ℝ^2$ $f(kx)=k^2f(x)$ for $k∈ℝ$ Show that any periodic solution of $y′=f(y)$ is constant. My attempt : Let $\lambda \in \mathbb{R}$. Let $g$ a periodic solution ...
7
votes
1answer
316 views

ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
5
votes
1answer
75 views

projectile motion (with height) complicated

When a child standing on a horizontal path throws a ball, it leaves her hand from a point that is 90cm vertically above the path. The ball clears a 4.5 m high wall that is 10.5 m away from ...
5
votes
1answer
85 views

Decomposite a vector field into two parts

Let A be a region in $\mathbb R^3$, and suppose $ \vec {\mathbf F}$ is a smooth vector field on A. I was asked to show that I can write $\vec {\mathbf F}=\vec {\mathbf F_1}+\vec {\mathbf F_2}$, s.t. ...
5
votes
2answers
700 views

General solution for $y^{iv}+ 2y''+y=\cos x$

Here is another problem from Mathews and Walker that has given me some trouble. 1-18. Find the general solution of $y^{iv}+ 2y''+y=\cos x$. Note: Thanks, everyone, for clearing up the ...
5
votes
0answers
2k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
4
votes
1answer
164 views

Some Scaling Estimate for Heat Kernel

NOTE. I have rewritten the question to summarize my current progress on this question. The bounty is for completing what I have done so far, or by offering a more elegant solution probably based on ...
4
votes
1answer
410 views

The characteristic polynomial of a recurrence relation.

If I have a linear homogeneous recurrence relation $$y_n=c_1y_{n-1}+\ldots+c_ky_{n-k},$$ I can get its characteristic equation, which is $$r^k=c_1r^{k-1}+\ldots+c_k.$$ In particular for ...
4
votes
1answer
101 views

Special Differential Equation (continued-2)

May you help me out in solving inhomogeneous differential equation looking like[this is radial part of Schrodinger equation]: ...
4
votes
4answers
1k views

Help with solving differential Equation using Exact Equation method

I need to learn how to solve differential equations using either the Exact Equation Approach and or the Special Integrating Factor methods. Below is a differential Equation to solve. $(2xy^2 + \cos ...
3
votes
3answers
149 views

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$

For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$. $k > 0$ is some constant. And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$. At ...
3
votes
1answer
2k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set ...
3
votes
1answer
441 views

Explain the error term in Euler method

Task: I had to find out some estimates for M and L to make sure the proportional accucrazy is not above $10^{-4}$ in the Euler method with the problem below. I am trying to understand the page 672 on ...
3
votes
4answers
1k views

particular solution of $4y''-y= \sin(x)\cdot \cos(x/2)$

So I'm working with a nonhomogeneous second order differential equation: $$4y''-y=\sin(x)\cos(x/2).$$ I know that the general solution, $y$, equals $y_c + y_p$ where $y_c$ is the general ...
2
votes
1answer
77 views

Stability analysis for a system of two differential equations

I have this system of differential equations: \begin{equation} \frac{dx}{dt}=\alpha x-\beta xy\\ \frac{dy}{dt}=\beta xy-\gamma y \end{equation} I need to find the critical points and then do a ...
2
votes
0answers
103 views

perturbation question

I'm a little stuck with a problem and I was hoping that you guys could help. Question: A projectile is fired up from the earth with an initial velocity of $v_0$ upwards. Accounting for air resistance, ...
2
votes
2answers
247 views

How to get the linear equation system for finite element method from the variational formulation

Let the problem be $$-u'' + a(x) u = f , \;x \in \Omega = ]0,1[ , \;u(0) = \alpha ; \;u(1) = \beta,$$ where $f \in L^2(\Omega) , a(x) \geq a_0 > 0 , a(x) \in L^{\infty}(\Omega).$ This problem ...
2
votes
2answers
137 views

Stability analysis, or, Can we prove this limit to be zero?

Let's think about this ODE $$ \dot{y}(t) = \gamma \left(g(t) - y(t)\right),\quad \gamma > 0, $$ where $g(t)$ is a Lipschitz continuous function. It can be seen that the value of $y(\cdot)$ goes ...
2
votes
2answers
1k views

second derivative of the inverse function

I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$ But how to derive the formula for the second derivative of g(y) knowing that $\left[\frac{1}{f(x)}\right]' = ...
2
votes
3answers
1k views

Cat Dog problem using integration

Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation ...
1
vote
1answer
24 views

Converting an ODE in polar form

Convert the ODE system $$ \dot{x}=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ into polar form. You should get two equations $$ \frac{d}{dt}\Phi(t)=...\\ ...
1
vote
1answer
82 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
1
vote
1answer
61 views

Linear equation solving

I cant solve $tx'+\dfrac{tx}{\sqrt{1+t^3}}=1$ I have tried to do it like an homogenian but i cant integrate $\dfrac{1}{\sqrt{1+t^3}}$ so i suposse it must be done by another method
1
vote
2answers
270 views

Comparison theorem for systems of ODE

Let vector-function $x(t)$ satisfy a differential equation $$ \dot x = f(x), $$ and a vector-function $y($t) satisfy a differential inequality $$ \dot y \leq f(y) $$ with starting positions $y(0) ...
1
vote
2answers
120 views

Easy way to solve this non-linear second degree DY $\sqrt{y}\;y''=1$?

I prooceeded by integrating both sides $$y'=\int y^{-\frac{1}{2}} dx=\cdots$$ so I got $(y')^{2}+C y' - \frac{1}{2} \sqrt{y} = 0$ but I am thinking that I am proceeding the wrong or the ...
1
vote
1answer
178 views

Help on solving an apparently simple differential equation

I need help to find an analytical solution to: $$p''(x)-k_1xp'(x)-k_2p(x)=0 \text{ where } k_1,k_2\in\mathbb R^+$$ with boundary conditions $p'(0)=0, p(r)=p(-r)=k_3$ where ...
7
votes
1answer
144 views

Solutions for $ \frac{dy}{dx}=y $?

Al-right, this may be a very basic question but I'm confused about this. We all know that one differential equation can only have one solution. Consider: $$ \frac{dy}{dx}=y $$ The solution is: $$ ...
7
votes
1answer
4k views

General Solution of a Differential Equation using Green's Function

My father recently lent me an old textbook of his, called Mathematical Methods of Physics by Mathews and Walker. I am working on the following exercise. Consider the differential equation ...
7
votes
1answer
228 views

Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...
6
votes
3answers
173 views

What went wrong?

Intrigued by this question, one-dimensional inverse square laws, I started to try to find an answer and came up with what follows. However, I calculated the derivatives to double check myself, and ...
6
votes
0answers
393 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
6
votes
2answers
186 views

Estimating rate of blow up of an ODE

Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$. Several related questions: (1) Can I ...
6
votes
5answers
1k views

Modelling forces acting on a sail

I'd like to create a basic model of the forces acting on a sail (wind sail, like a tail ship) A couple of things I was thinking about: 1) can create a very simple model where wind is 'one' force ...
5
votes
1answer
89 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?

Can an arbitrary function $ y: \mathbb{R} \to \mathbb{R} $ always be expressed as $ \dfrac{z'}{z} $ for some differentiable function $ z: \mathbb{R} \to \mathbb{R} $, or are additional conditions on $ ...
5
votes
0answers
268 views

Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com This is a pet project that I plan to use to convince my Prof that I would rather try something similar to this than to do the prescribed project. Edit : ...
5
votes
1answer
41 views

Can a nice enough ODE always be extended to the complex plane?

Suppose I have a first-order ODE $y' = f(x, y)$, where $y: \mathbb{R} \to \mathbb{R}$, and $f \in \mathbb{R}[x, y]$. Consider $f^\mathbb{C} = i(f)$, where $i: \mathbb{R}[x, y] \to \mathbb{C}[x, y]$ is ...
5
votes
2answers
305 views

Legendre polynomials, Laguerre polynomials: Basic concept

I am asking a simple conceptual question. I saw in many Mathematics and Mathematical physics text books that the Legendre polynomials and Laguerre polynomials "falling from the sky"! I mean, I didn't ...
5
votes
0answers
108 views

Uniqueness result in linear differential equation of degree $n$.

Suppose that $f$ is such that $$f^{(n)}=\sum_{j=0}^{n-1}a_jf^{(j)}$$ Some little work is needed to get to ($a_j=0$ if $j<0$) $${f^{(n + 1)}} = \sum\limits_{j = 0}^{n - 1} {\left( {{a_{j - 1}} + ...